There are a lot of reasons to love Lily. Here are a few.
I love Lily because she stuck up for her friends. It didn't matter if she was the only one doing it. It didn't matter if their tormentor was powerful or popular. She was powerful, too, and she had no compunctions about using that power to protect the people she cared about.
I love Lily because she was bold. She spoke her mind to both teachers and students, be they Head of House or the most popular in their year. She wasn't easily impressed and she was even less easily cowed.
I love Lily because she wasn't perfect. She loved her sister, but she snooped in her things. She snapped back with hurtful taunts when Petunia was cruel to her. She certainly wasn't above name-calling.
I love Lily because she wasn't afraid to confront her friends when she was angry with them, or afraid for them. She wasn't even afraid to lose them if they've hurt her too much, if they've pushed her too far. No one is obligated to stay friends with someone who has hurt them. Lily knew that, and she was better off for it.
I love Lily because she did give people second chances. She gave Severus a number of them, until he took one too many - because he was her friend, because she saw beyond his faults. She gave James a chance to prove he was more than an arrogant bully, that he could grow up and be a better man.
I love Lily because she was brave. It would have been easier for her to hide from a war that was so much more dangerous for people like her. Instead, she fought. She and James faced Voldemort three times.
I love Lily because she was compassionate. She felt for her sister. She defended Remus when she barely knew him. She was angry over people she didn't know being hurt. She cried all evening over the McKinnons.
I love Lily because of her sacrifice. Steadfast courage and powerful love in the face of certain death.
I love Lily because she was, I think, the best a Gryffindor could be.